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Data Interview Question

Winning in a Coin Toss Game

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Solution & Explanation

The problem presented is a classic example of a stochastic process known as the Gambler's Ruin problem. In this scenario, a gambler starts with a certain amount of money and continues to gamble until they either lose all their money or reach a target amount. The key here is to determine the probability of the gambler reaching the target amount before going broke.

Problem Breakdown:

  • Initial Capital: $30
  • Target Capital: $100
  • Bankruptcy Point: $0
  • Outcome of Each Game: Win 1forheads,lose1 for heads, lose 1 for tails.
  • Probability of Heads or Tails: 0.5 each (assuming a fair coin).

Approach:

  1. Understanding the Random Walk:

    • Each coin flip represents a step in a random walk.
    • The gambler's current capital can increase by 1(forheads)ordecreaseby1 (for heads) or decrease by 1 (for tails).
    • The process continues until the gambler reaches 100or100 or 0.
  2. Expected Value and Probability:

    • The expected value of the gambler's capital remains constant at $30 throughout the game. This is because the expected change in capital per flip is zero (0.5 * 1 + 0.5 * (-1) = 0).
    • The expected value at the end of the game is a weighted sum of the possible outcomes (100or100 or 0) multiplied by their respective probabilities.
  3. Setting Up the Equation:

    • Let r be the probability of reaching $100.

    • The probability of reaching $0 is 1 - r.

    • The expected value equation can be set up as follows:

      E[X]=100×r+0×(1r)=100rE[X] = 100 \times r + 0 \times (1-r) = 100r

      Since we know that the expected value is $30, we substitute:

      30=100r30 = 100r

      Solving for r gives us:

      r=30100=0.3r = \frac{30}{100} = 0.3

  4. Conclusion:

    • The probability that the gambler will reach $100 before going broke is 0.3, or 30%.

Additional Insights:

  • Markov Chain Perspective:

    • This problem can also be approached using Markov chains, where each state represents a possible amount of money the gambler can have.
    • The transition probabilities between states are determined by the outcomes of the coin toss.
  • Symmetry and Fairness:

    • The fairness of the coin ensures that the expected value remains constant, highlighting the inherent symmetry in the problem.
  • Practical Implications:

    • In real-world scenarios, understanding such probability models can assist in decision-making processes involving risk and reward, such as financial investments or strategic games.

This explanation provides a comprehensive understanding of the problem and its solution, integrating concepts from probability theory and stochastic processes.